Of the two cartridges that you mentioned, the .30-06 would have a slight edge over the .308, especially in a modern scoped rifle. With proper bullets and shot placement, either would kill and of the animals that you listed.
Historically, bolt actions were stronger and more accurate than other actions.
Although I have a couple of traditional iron sighted muzzleloading rifles and two Model 94 Winchesters, I personally prefer a clean bolt action rifle with a scope and no iron sights.
Another benefit of scopes is as we get older, it becomes harder to see iron sights. When scopes are properly adjusted, the cross hairs and the target are in focus at any range.
My current favorite "all around" rifle is my .300 Weatherby vanguard. It has served me very well on a number of North American and African hunts.